The Nash-Moser embedding [1]} gives an isometric embedding
\(j\colon \left(M, \, g\right)\rightarrow \left(\mathbb {R}^{\ell }, \, (\cdot ,\cdot )_{\mathbb {R}^\ell }\right)\) . Therefore, we define
\(H^1(a, \, b; \, M):=\left\lbrace v\in H^1(a, \, b; \, \mathbb {R}^\ell ): v(t)\in j(M)\quad \text{ for any }t\in (a,b)\right\rbrace \) 
